January 2016 - mathfun@mailman.xmission.com

[math-fun] 3D version of central angle theorem
by James Propp 05 Jan '16

05 Jan '16

Is there a version of the central angle theorem that applies to spheres and solid angles? The central angle theorem (in one formulation) tells us that if C is a circle containing the point p and C' is a circle centered at p, then projection from C to C' through p halves all angles. if S is a sphere containing the point p and S' is a sphere centered at p, does projection from S to S' through p halve all solid angles? By the way, I spent about three minutes fruitlessly searching the internet for an answer, and I figure that that's enough, since one of you has probably thought about this before and can answer my question in under three minutes. But if some of you think that I (and other posters) should do more googling before we post our questions, please let me know. Jim Propp

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[math-fun] Saddle-shaped minimal surface bounding a non-planar hexagon
by James Propp 04 Jan '16

04 Jan '16

Do any of you know of any pictures (hand-drawn or computer-generated) of the saddle-shaped surface you get when you make a non-planar hexagonal frame consisting of all the edges of a cube that avoid two antipodal vertices an dip it in a soap film solution? This is not to be confused with the surface you get when you dip a non-planar octagonal frame consisting of the edges in a Hamiltonian cycle on the cube. I spent about five minutes searching images.google.com and didn't find what I'm looking for, and would appreciate help from any of you who may know about where such things can be found! Thanks, Jim Propp

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[math-fun] "Holy grail" proportional representation election method
by Warren D Smith 04 Jan '16

04 Jan '16

Forest Simmons and I finally found the long-sought "holy grail" combination of properties for a multiwinner election method. (Unless we didn't, because my proof has some bug...) I had previously conjectured the holy grail was impossible. If anybody wants to read preliminary manuscript explaining: http://rangevoting.org/HolyGrailPR.html Unfortunately, our holy grail election method is computationally hard, large computer runtime. I do not know whether that is necessary. I proved that PR (proportional representation multiwinner) elections, to assure good quality results, inherently must solve NP-hard problems. That kind of hardness does not bother me since it is unavoidable. But the holy grail method we discovered is worse than that, in the sense that the "optimum" parliament which it finds, i.e. the one with maximum quality... well, now the quality function ITSELF seems hard to evaluate. I may be able (although so far I haven't) to prove my quality function is NP-hard to evaluate. Is there a way to attain the holy grail with a polytime quality function? We presently do not know, but that would make me a lot happier. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] q^9 in a theta constant
by Bill Gosper 03 Jan '16

03 Jan '16

https://en.wikipedia.org/wiki/Dedekind_eta_function gives a peculiar theta expression for Dedekind eta. Combining with the more obvious DedekindEta[\[Tau]] -> EllipticTheta[1, Pi/3, E^((I Pi \[Tau])/3)]/Sqrt[3] gives the weird-looking EllipticTheta[1,Pi/3, q] == -Sqrt[3] q^(25/4) EllipticTheta[4, 15/2 I Log[q], q^9] == (Sqrt[Pi/3] EllipticTheta[1, Pi/3, E^(Pi^2/(9 Log[q]))])/Sqrt[-Log[q]] (The latter via Jacobi's imaginary transformation.) --rwg

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